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556 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000
Fair End-to-End Window-Based Congestion ControlJeonghoon Mo and Jean Walrand , Fellow, IEEE
Abstract—In this paper, we demonstrate the existence of fairend-to-end window-based congestion control protocols for packet-switched networks with first come–first served routers. Our defini-tion of fairness generalizes proportional fairness and includes ar-bitrarily close approximations of max–min fairness. The protocolsuse only information that is available to end hosts and are designedto converge reasonably fast.
Our study is based on a multiclass fluid model of the network.The convergence of the protocols is proved using a Lyapunov func-tion. The technical challenge is in the practical implementation of the protocols.
Index Terms—Bandwidth sharing, congestion control, fairness,TCP, window.
I. INTRODUCTION
WE STUDY the existence of fair end-to-end congestion
control schemes. More precisely, the question is that of
the existence of congestion control protocols that converge to a
fair equilibrium without the help of the internal network nodes,
or routers. Using such a protocol, end-nodes, or hosts, monitor
their connections. By so doing, the hosts get implicit feedback
from the network such as round-trip delays and throughput but
no explicit signals from the network routers. The hosts imple-
ment a window congestion control mechanism. Such end-to-end
control schemes do not need any network configuration and
therefore could be implemented in the Internet without modi-
fying the existing routers or the IP protocol.
The Internet congestion control is implemented in end-to-end
protocols. The motivation for such protocols is that they place
the complex functions in the hosts and not inside the network.
Consequently, only the hosts that want to implement different
complex functions need to have their software upgraded. An-
other justification, which is more difficult to make precise, is
that by keeping the network simple it can scale more easily.
TCP is the most widely used end-to-end protocol in the In-
ternet. When using TCP [12], a source host adjusts its windows
size, the maximum amount of outstanding packets it can send
to the network, to avoid overloading routers in the network and
the destination host.
Many researchers have observed that, when using TCP, con-nections with a long round-trip time that go through many bot-
tlenecks have a smaller transmission rate than the other con-
nections [4], [6], [22]. To improve fairness, Floyd and Jacobson
Manuscript received November 10, 1998; revised January 19, 1999 and Oc-tober 4, 1999; approvedby IEEE/ACM TRANSACTIONSON NETWORKING EditorS. Floyd. This work was supported in part by a grant from the National ScienceFoundation.
J. Mo is with AT&T Labs, Middletown, NJ 07748 USA (e-mail: [email protected]).
J. Walrand is with the University of California, Berkeley, CA 94704 USA.Publisher Item Identifier S 1063-6692(00)09125-1.
[5] proposed a “constant rate adjustment” algorithm and Han-
derson et al. [9] simulated a variation of this scheme. How-
ever, choosing the parameters of such algorithms is still an open
problem.Thus, although end-to-end protocols such as those im-
plemented in TCP are very desirable for extensibility and
scalability reasons, they are unfair. Roughly, a fair scheme
is one that does not penalize some users arbitrarily. Accord-
ingly, the question that arises naturally is the existence of fair
end-to-end congestion protocols.
In an early paper, Jaffe [14] shows that power, defined as
, cannot be optimized in a dis-
tributed manner.
Chiu and Jain [3] show that in a network with users thatshare a unique bottleneck node, a linear increase and multiplica-
tive decrease algorithm converges to an efficient and fair equi-
librium. Most current implementations of TCP window-based
control use a linear increase and multiplicative decrease of the
window size, as suggested in [12]. However, these implemen-
tations control the size of their window and not their transmis-
sion rate. Moreover, simple examples show that the algorithm
does not converge to an efficient and fair equilibrium in net-
works with multiple bottleneck nodes.
Shenker [28] considers a limited class of protocols and ar-
gues that “no aggregate feedback control is guaranteed fair.”
This statement suggests that end-to-end control cannot guar-
antee convergence to a fair equilibrium. Unfortunately, the classof protocols that he considers excludes many implementable
end-to-end protocols. Jain and Charny refers to [28] to justify
the necessity of switch-based control for fairness [15], [2].
Recently, Kelly et al. [18] proposed proportional fairness and
exhibited an aggregate feedback algorithm that converges to the
point. In their scheme, each user is implementing a linear in-
crease and multiplicative decrease of its rate based on an ad-
ditive feedback from the routers the connection goes through.
This protocol requires that the routers can signal the difference
between their load and their capacity. In our protocol, each host
controls its window size not its rate, based on the total delay.
Our protocol can be viewed as a refinement of TCP congestion
control algorithms.Le Boudec et al. [10], [21] studied the exact form of fair-
ness founded by the linear increase, multiplicative decrease al-
gorithms. Massoulie and Roberts have done interesting work on
bandwidth sharing which is similar to ours [23]. They proposed
fixed window algorithms to achieve fairness.
In this paper, we revisit the fundamental question of the ex-
istence of fair end-to-end protocols and we provide a positive
answer by constructing explicitly such protocols. The rest of the
paper is organized as follows. In Section II, we present a multi-
class fluid model for window-based control and theoretical re-
1063–6692/00$10.00 © 2000 IEEE
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564 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 8, NO. 5, OCTOBER 2000
TABLE ITHROUGHPUT OF 5 CONNECTION
TABLE IITHROUGHPUT DIFFERENTIATION WITH DIFFERENT TARGETS
Fig. 4. Plot of window and packets with p = 2, 6, 10, 14, and 18.
propagation delays to show that our algorithm does not suffer
from round-trip delay bias.
B. Fairness
We ran simulations for five connections with of two
packets for 100 s. The packet size is 1 KB. We measure the
throughput achieved by those five connections. Table I shows
the results. The second and third columns show the throughputs
of connections when -fair algorithms and TCP-Renos
are used respectively. It can be observed that our algorithm
achieves fairer throughput without regard to the delay while the
throughput of TCP-Reno varies more.
We use a different value of for each connection. Table II
shows the throughput of each connections with 2, 6, 10,
14, and 18 packets for 60 s.The buffer size is of the router is 100 packets and the
propagation delay of each connection is now set to 3 ms.
The left-hand side of Fig. 4 plots the window size over time
of the five connections. The bottom line corresponds to the con-
nection with target equal to 2 and the top line to that with target
equal to 18. Note that a bigger target results in a bigger window
size.
The right-hand side of Fig. 4 and Table II show that the
throughput of connection is almost proportional to . The
second column in Table II is the number of packets acknowl-
edged for the five connections during the last 50 s. We omit
the first 10 s to remove the effect of the slow start. Comparing
the third column with the fourth shows that throughput is
almost proportional to the target. This simulations shows that
by controlling the target backlog ( ), we can control the
throughput of each connection.
VI. CONCLUSION
In this paper, we have addressed the fundamental question
of the existence of fair end-to-end window-based congestion
control. We have shown the existence of window-based fair
end-to-end congestion control using a multiclass closed fluid
model. We showed that the flow rate vector is a well-defined
function of the window-size vector and characterized this func-
tion. We generalized the proportional fairness and related the
fairness to the optimization problem. Our definition of fairness
addresses the compromise between user fairness and resource
utilization. With the help of an optimization problem, we have
related window sizes and the fair rates. We have developed an
algorithm which converges to the fair rates and proved its con-
vergence using a Lyapunov function.
Our algorithm uses the propagation delay , measured delay, and window size . Unfortunately, the end user cannot
know the exact value of propagation delay. Furthermore, the
value of propagation delay could change in the case of rerouting
in packet-switched networks. TCP-Vegas uses the minimum of
delays observed so far as an estimated propagation delay. TCP-
Vegas fails to adapt to the route change when the changed route
is longer than original route. Refer to La et al. [26], [19] for this
problem. Theimpact of feedback delays on the convergence will
be also be topic of further studies.
APPENDIX AUNIQUENESS OF THE RATE VECTOR
Theorem 2: Given , the flow rate vector satis-
fying (1)–(4) is unique. We use two lemmas in the proof of the
Theorem 2. The first one is a partial result of Rosen [27] and the
second is Farkas’ lemma (see, e.g., [24]).
Lemma 5: Let be a vector of real-valued
functions defined on . If the Jacobian matrix exists
and is either positive definite for all or negative definite
for all , then there is at most one such that
holds.
Proof: Assume there are two distinct points and
such that for . Let
for . Since is the Jacobian of , we have
Hence
Multiplying both sides by gives
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MO AND WALRAND: FAIR END-TO-END WINDOW-BASED CONGESTION CONTROL 567
Now, (45) implies that . Hence
(48)
Plugging (48) into (46) gives (41).
ACKNOWLEDGMENT
The authors would like to thank J.-Y. Le Boudec for his
helpful criticism and give credit to him for the proof of
Lemma 3. The authors also express their gratitude to R. La
for helping to complete the work. Finally, they thank S. Floyd,
S. Low, and the anonymous reviewers for their comments to
improve the paper.
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Jeonghoon Mo received the B.S. and M.S. degreesfrom Seoul National University, Seoul, Korea. He re-ceived the M.S. and Ph.D. degrees from the Univer-sity of California, Berkeley.
Currently, he is a Senior Technical Staff Memberwith AT&T Labs, Middletown, NJ. His researchinterests include communication networks, queueingtheory, Internet protocols, and quality-of-serviceissues.
Jean Walrand (S’71–M’80–SM’90–F’93) is aProfessor in the Department of Electrical Engi-neering and Computer Sciences at the Universityof California, Berkeley. His research interestsinclude stochastic processes, queueing theory,communication networks, and control systems. He isthe author of An Introduction to Queueing Networks(Prentice Hall, 1988), Communication Networks: AFirst Course (McGraw-Hill, 1998), and co-authorof High-Performance Communication Networks(Morgan Kaufmann, 2000).
Dr. Walrand is a recipient of the Lanchester Prize from the Operations Re-search Society of America and of the S. Rice Prize from the Communication
Society of IEEE.